Quantum mechanics in the general quantum systems (V): Hamiltonian eigenvalues

TL;DR

This paper presents an exact series expression for Hamiltonian eigenvalues in general quantum systems, along with a novel algebraic approach for calculating these eigenvalues, validated by application to a quartic anharmonic oscillator.

Contribution

It introduces a complete, non-approximate series expression for Hamiltonian eigenvalues and a new algebraic method for their calculation in arbitrary quantum systems.

Findings

Derived a complete series expression for eigenvalues.

Proposed an algebraic approach involving a kernel function.

Validated method by calculating ground state energy of a quartic oscillator.

Abstract

We derive out a complete series expression of Hamiltonian eigenvalues without any approximation and cut in the general quantum systems based on Wang's formal framework \cite{wang1}. In particular, we then propose a calculating approach of eigenvalues of arbitrary Hamiltonian via solving an algebra equation satisfied by a kernal function, which involves the contributions from all order perturbations. In order to verify the validity of our expressions and reveal the power of our approach, we calculate the ground state energy of a quartic anharmonic oscillator and have obtained good enough results comparing with the known one.